Quantitative logarithmic equidistribution of the crucial measures

Quantitative logarithmic equidistribution of the crucial measures
Jacobs, Kenneth
2018-02-08 00:00:00
Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value, and let
$$\phi \in K(z)$$
ϕ
∈
K
(
z
)
with
$$\deg (\phi )\ge 2$$
deg
(
ϕ
)
≥
2
. Recently, Rumely introduced a family of discrete probability measures
$$\{\nu _{\phi ^n}\}$$
{
ν
ϕ
n
}
on the Berkovich line
$$\mathbf{P }^1_{\text {K}}$$
P
K
1
over K which carry information about the reduction of conjugates of
$$\phi $$
ϕ
. In a previous article, the author showed that the measures
$$\nu _{\phi ^n}$$
ν
ϕ
n
converge weakly to the canonical measure
$$\mu _\phi $$
μ
ϕ
. In this article, we extend this result to allow test functions which may have logarithmic singularities at the boundary of
$$\mathbf{P }^1_{\text {K}}$$
P
K
1
. These integrands play a key role in potential theory, and we apply our main results to show the potential functions attached to
$$\nu _{\phi ^n}$$
ν
ϕ
n
converge to the potential function attached to
$$\mu _\phi $$
μ
ϕ
, as well as an approximation result for the Lyapunov exponent of
$$\phi $$
ϕ
.
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pngResearch in Number TheorySpringer Journalshttp://www.deepdyve.com/lp/springer-journals/quantitative-logarithmic-equidistribution-of-the-crucial-measures-dpNsLtmdXL

Abstract

Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value, and let
$$\phi \in K(z)$$
ϕ
∈
K
(
z
)
with
$$\deg (\phi )\ge 2$$
deg
(
ϕ
)
≥
2
. Recently, Rumely introduced a family of discrete probability measures
$$\{\nu _{\phi ^n}\}$$
{
ν
ϕ
n
}
on the Berkovich line
$$\mathbf{P }^1_{\text {K}}$$
P
K
1
over K which carry information about the reduction of conjugates of
$$\phi $$
ϕ
. In a previous article, the author showed that the measures
$$\nu _{\phi ^n}$$
ν
ϕ
n
converge weakly to the canonical measure
$$\mu _\phi $$
μ
ϕ
. In this article, we extend this result to allow test functions which may have logarithmic singularities at the boundary of
$$\mathbf{P }^1_{\text {K}}$$
P
K
1
. These integrands play a key role in potential theory, and we apply our main results to show the potential functions attached to
$$\nu _{\phi ^n}$$
ν
ϕ
n
converge to the potential function attached to
$$\mu _\phi $$
μ
ϕ
, as well as an approximation result for the Lyapunov exponent of
$$\phi $$
ϕ
.